How can I take an image and create a flat 3D model out of it?
It's absolutely possible, look up billboard sprites. For generating them in Blender, get a render with transparency enabled (more googling) of one tree and then you can duplicate it around to fake a forest. It's also potentially a good idea to render some far off trees to make a distant background rather than rendering potentially hundreds of those billboard sprites.  It might be fun to experiment with more roomy tree renderings given you don't have to have unity do that part anymore...

Edit: totally opposite approach is what's being asked for here!

Instead, you can also start from the geometry which was alpha masked, and subdivide it (use simple instead of catmul-clark) then select verts that are outside the texture map and delete them. Probably don't want to subdivide too far before doing this because deleting verts can get boring...